Factorize the following : (x²-6xy + y²) – 25

Q: Factorize the following : (x²-6xy + y²) – 25

To factorize the expression $(x^2 - 6xy + y^2) - 25$, we need to recognize the structure of the expression and apply appropriate algebraic identities. 1. **Recognizing the Structure:** The expression $(x^2 - 6xy + y^2)$ can be rewritten as $(x - y)^2$. This is because: \[ (x - y)^2 = x^2 - 2xy + y^2 \] However, our expression is $x^2 - 6xy + y^2$, which can be rewritten as: \[ (x - 3y)^2 = x^2 - 6xy + 9y^2 \] Therefore, the expression becomes: \[ (x - 3y)^2 - 25 \] 2. **Applying the Difference of Squares:** The expression $(x - 3y)^2 - 25$ is a difference of squares, which can be factored using the identity $a^2 - b^2 = (a - b)(a + b)$. Here, $a = (x - 3y)$ and $b = 5$. Thus, we have: \[ (x - 3y)^2 - 5^2 = ((x - 3y) - 5)((x - 3y) + 5) \] Simplifying, we get: \[ (x - 3y - 5)(x - 3y + 5) \] 3. **Verification of Option 4:** The factorization $(x - 3y - 5)(x - 3y + 5)$ matches Option 4 exactly. Therefore, Option 4 is the correct factorization of the given expression. 4. **Explanation of Incorrect Options:** - **Option 1: $(x - 5)(x + 5)$** - This option represents the factorization of $x^2 - 25$, which is not applicable to our expression since it does not account for the $y$ terms. - **Option 2: $(x - y - 5)(x - y + 5)$** - This option represents the factorization of $(x - y)^2 - 25$, which does not match our expression because it simplifies $x^2 - 2xy + y^2 - 25$, not $x^2 - 6xy + y^2 - 25$. - **Option 3: $(x + 3y + 5)(x - 3y - 5)$** - This option does not correctly factor the expression because it introduces terms that do not simplify to the original expression $(x^2 - 6xy + y^2) - 25$. In conclusion, Option 4 is the correct factorization of the given expression, as it accurately applies the difference of squares to $(x - 3y)^2 - 25$.

Question

Accepted Answer

To factorize the expression $(x^2 - 6xy + y^2) - 25$, we need to recognize the structure of the expression and apply appropriate algebraic identities.

1. **Recognizing the Structure:**
   The expression $(x^2 - 6xy + y^2)$ can be rewritten as $(x - y)^2$. This is because:
   $$
   (x - y)^2 = x^2 - 2xy + y^2
   $$
   However, our expression is $x^2 - 6xy + y^2$, which can be rewritten as:
   $$
   (x - 3y)^2 = x^2 - 6xy + 9y^2
   $$
   Therefore, the expression becomes:
   $$
   (x - 3y)^2 - 25
   $$

2. **Applying the Difference of Squares:**
   The expression $(x - 3y)^2 - 25$ is a difference of squares, which can be factored using the identity $a^2 - b^2 = (a - b)(a + b)$. Here, $a = (x - 3y)$ and $b = 5$. Thus, we have:
   $$
   (x - 3y)^2 - 5^2 = ((x - 3y) - 5)((x - 3y) + 5)
   $$
   Simplifying, we get:
   $$
   (x - 3y - 5)(x - 3y + 5)
   $$

3. **Verification of Option 4:**
   The factorization $(x - 3y - 5)(x - 3y + 5)$ matches Option 4 exactly. Therefore, Option 4 is the correct factorization of the given expression.

4. **Explanation of Incorrect Options:**
   - **Option 1: $(x - 5)(x + 5)$**
     - This option represents the factorization of $x^2 - 25$, which is not applicable to our expression since it does not account for the $y$ terms.
   
   - **Option 2: $(x - y - 5)(x - y + 5)$**
     - This option represents the factorization of $(x - y)^2 - 25$, which does not match our expression because it simplifies $x^2 - 2xy + y^2 - 25$, not $x^2 - 6xy + y^2 - 25$.
   
   - **Option 3: $(x + 3y + 5)(x - 3y - 5)$**
     - This option does not correctly factor the expression because it introduces terms that do not simplify to the original expression $(x^2 - 6xy + y^2) - 25$.

In conclusion, Option 4 is the correct factorization of the given expression, as it accurately applies the difference of squares to $(x - 3y)^2 - 25$.